16 Anisotropic Extension of the Exchange Factor Transformation

16.1 Introduction

The exchange factor transformation described in Chapter Chapter 9 enables analytical treatment of multiple reflection-scattering processes in radiative transfer by transforming the exchange factor matrix \mathbf{F} into absorption and reflection-scattering matrices \mathbf{A} and \mathbf{R}. The method assumes isotropic reflection-scattering behavior, where the probability of reflection-scattering from element j is independent of the incident element i. This is represented by a column-constant single reflection-scattering matrix \mathbf{B}.

However, many physical systems exhibit anisotropic reflection-scattering, where the probability and directional distribution of reflection-scattering depend on the incident direction. This chapter presents an extension of the exchange factor transformation that accommodates anisotropic reflection-scattering while preserving all fundamental properties: non-negativity, energy conservation, and analytical solvability.

16.2 Anisotropic Reflection-Scattering Matrix

The anisotropic extension introduces a full matrix \mathbf{\Gamma} (Gamma) to characterize directional dependence of reflection-scattering:

\mathbf{\Gamma} \in \mathbb{R}^{N \times N}, \quad 0 \leq \mathbf{\Gamma}_{ij} \leq 1 \quad \forall \; ij \tag{16.1}

where \mathbf{\Gamma}_{ij} represents the anisotropy parameter controlling how incident radiation from element i affects reflection-scattering from element j.

16.2.1 Physical Interpretation

The matrix \mathbf{\Gamma} provides directional control over reflection-scattering behavior:

\mathbf{\Gamma}_{ij} = 0.5: Baseline (isotropic) reflection-scattering from j due to incidence from i
\mathbf{\Gamma}_{ij} < 0.5: Reduced reflection-scattering from j due to incidence from i (directional suppression)
\mathbf{\Gamma}_{ij} > 0.5: Enhanced reflection-scattering from j due to incidence from i (directional enhancement)

When \mathbf{\Gamma}_{ij} = 0.5 for all elements (or equivalently when the scaling coefficient c = 0), the anisotropic formulation reduces exactly to the isotropic case with column-constant reflection-scattering matrix \mathbf{B}.

16.3 Construction of the Anisotropic Interaction Matrix

The anisotropic interaction-reflection-scattering matrix \mathbf{K} is constructed through a multi-step process that ensures physical consistency.

Step 1: Initial Reflection-Scattering Distribution

Define the initial reflection-scattering distribution matrix:

\mathbf{B}_{\mathrm{init},ij} = \mathbf{b}_j + c(2\mathbf{\Gamma}_{ij} - 1)\min(\mathbf{b}_j, 1-\mathbf{b}_j) \tag{16.2}

where:

\mathbf{b}_j is the baseline reflection-scattering coefficient of element j
\mathbf{\Gamma}_{ij} \in [0,1] controls directional anisotropy
c > 0 is a scaling coefficient that determines the strength of anisotropic perturbations
\min(\mathbf{b}_j, 1-\mathbf{b}_j) limits the perturbation range to maintain physical validity

This parameterization maps:

\mathbf{\Gamma}_{ij} = 0 \rightarrow maximum downward perturbation: \mathbf{B}_{\mathrm{init},ij} = \mathbf{b}_j - c\min(\mathbf{b}_j, 1-\mathbf{b}_j)
\mathbf{\Gamma}_{ij} = 0.5 \rightarrow baseline (isotropic): \mathbf{B}_{\mathrm{init},ij} = \mathbf{b}_j
\mathbf{\Gamma}_{ij} = 1 \rightarrow maximum upward perturbation: \mathbf{B}_{\mathrm{init},ij} = \mathbf{b}_j + c\min(\mathbf{b}_j, 1-\mathbf{b}_j)

The coefficient c must be chosen to guarantee that the final matrix \mathbf{K} satisfies 0 \leq \mathbf{K}_{ij} \leq \mathbf{F}_{ij} for all elements. The maximum safe value c_{\max} can be computed analytically from the constraint equations (see Chapter Chapter 19 for derivation). The maximum safe coefficient is:

c_{\max} = \min\left(c_{\max,\text{nonneg}}, c_{\max,\text{leq F}}\right)

where:

c_{\max,\text{nonneg}} = \min_{i,j : \text{denom} > 0} \frac{b_j}{\Delta_i(1-b_j) - (2\Gamma_{ij}-1)m_j}

c_{\max,\text{leq F}} = \min_{i,j : \text{denom} > 0} \frac{1-b_j}{(2\Gamma_{ij}-1)m_j - \Delta_i(2-b_j)}

For numerical safety, a lower value is used: c = \alpha \cdot c_{\max} with \alpha < 1.

This guarantees 0 \leq \mathbf{K}_{ij} \leq \mathbf{F}_{ij} for all elements.

Key Properties

Negative B allowed: \mathbf{B}_{\mathrm{init},ij} can be negative for sufficiently large c, but the transformation guarantees \mathbf{K}_{ij} \geq 0.
Element-wise freedom: Full N \times N degrees of freedom in \mathbf{B}_{\mathrm{init}}.
Optimal bounds: Since the bounds on c were derived analytically they provide maximum flexibility for any given \mathbf{\Gamma}.

Step 2: Interaction-Reflection-Scattering (Pre-Normalization)

Compute the element-wise product of the exchange factor matrix and the initial reflection-scattering distribution:

\mathbf{K}_{\mathrm{init},ij} = \mathbf{F}_{ij} \cdot \mathbf{B}_{\mathrm{init},ij} \tag{16.3}

This represents the probability that radiation emitted from element i interacts with element j and is then reflected-scattered, before normalization.

Step 3: Enforce Probability Conservation

Create a matrix that accounts for both reflection-scattering and absorption:

\mathbf{U}_{ij} = \mathbf{K}_{\mathrm{init},ij} + \mathbf{F}_{ij} \cdot \mathbf{P}_{jj} \tag{16.4}

Each emitted ray either reflects-scatters (\mathbf{K}_{\mathrm{init}}) or is absorbed (\mathbf{F} \cdot \mathbf{P}).

Step 4: Compute Row Sums

Calculate the row sums:

\mathbf{s}_i = \sum_j \mathbf{U}_{ij} = \sum_j (\mathbf{K}_{\mathrm{init},ij} + \mathbf{F}_{ij} \cdot \mathbf{P}_{jj}) \tag{16.5}

Step 5: Normalize to Row-Stochastic

Normalize the matrix to ensure row sums equal unity:

\mathbf{T}_{ij} = \frac{\mathbf{U}_{ij}}{\mathbf{s}_i} \tag{16.6}

This creates a row-stochastic matrix, ensuring that total probability is conserved for radiation emitted from each element i.

Step 6: Final Interaction-Reflection-Scattering Matrix

Extract the reflection-scattering component:

\mathbf{K}_{ij} = \mathbf{T}_{ij} - \mathbf{F}_{ij} \cdot \mathbf{P}_{jj} \tag{16.7}

Substituting from Eq. 16.6:

\mathbf{K}_{ij} = \frac{\mathbf{K}_{\mathrm{init},ij} + \mathbf{F}_{ij} \cdot \mathbf{P}_{jj}}{\mathbf{s}_i} - \mathbf{F}_{ij} \cdot \mathbf{P}_{jj} \tag{16.8}

Simplifying:

\mathbf{K}_{ij} = \frac{\mathbf{K}_{\mathrm{init},ij}}{\mathbf{s}_i} - \mathbf{F}_{ij} \cdot \mathbf{P}_{jj} \left(1 - \frac{1}{\mathbf{s}_i}\right) \tag{16.9}

16.4 Properties of the Row Sum Normalization Factor

16.4.1 Bounds on s

From Eq. 16.5:

\mathbf{s}_i = \sum_j (\mathbf{K}_{\mathrm{init},ij} + \mathbf{F}_{ij} \cdot \mathbf{P}_{jj})

Since \mathbf{K}_{\mathrm{init},ij} = \mathbf{F}_{ij} \cdot \mathbf{B}_{\mathrm{init},ij}, we have:

\mathbf{s}_i = \sum_j \mathbf{F}_{ij} \cdot (\mathbf{B}_{\mathrm{init},ij} + \mathbf{P}_{jj}) \tag{16.10}

Since \mathbf{F} is row-stochastic (\sum_j \mathbf{F}_{ij} = 1), \mathbf{F}_{ij} \geq 0, and \mathbf{P}_{jj} = 1-\mathbf{b}_j > 0:

\mathbf{s}_i > 0 \tag{16.11}

The normalization factor \mathbf{s}_i is always positive, preventing division by zero in equations Eq. 16.8 and Eq. 16.9.

16.4.2 Special Case: Isotropic Limit

When \mathbf{\Gamma}_{ij} = 0.5 for all i,j (or equivalently when c = 0), we have \mathbf{B}_{\mathrm{init},ij} = \mathbf{b}_j (column-constant):

\mathbf{s}_i = \sum_j \mathbf{F}_{ij} \cdot (\mathbf{b}_j + (1-\mathbf{b}_j)) = \sum_j \mathbf{F}_{ij} = 1

Therefore, the normalization does nothing, and:

\mathbf{K}_{ij} = \mathbf{F}_{ij} \cdot \mathbf{b}_j

This recovers the isotropic case exactly: \mathbf{K} = \mathbf{F} \circ \mathbf{B}, where \mathbf{B} is the column-constant matrix of reflection-scattering coefficients.

16.5 Subsequent Transformation

Once \mathbf{K} is constructed, the remainder of the exchange factor transformation proceeds almost identically to the isotropic case:

The steady-state path matrix is given by:

\mathbf{S}_\infty = (\mathbf{I} - \mathbf{K})^{-1} \mathbf{F} \tag{16.12}

The absorption and reflection-scattering matrices are given by the alternate formulation:

\mathbf{A} = \mathbf{P}(\mathbf{I}-\mathbf{K})^{-1}(\mathbf{F}-\mathbf{K}) \tag{16.13}

\mathbf{R} = \mathbf{P}(\mathbf{I}-\mathbf{K})^{-1}\mathbf{K} \tag{16.14}

The system matrice \mathbf{C} and \mathbf{D} are given by:

\mathbf{C} = \mathbf{I} - \mathbf{A}^T - \mathbf{R}^T \tag{16.15}

\mathbf{D} = \mathbf{I} - \mathbf{R}^T \tag{16.16}

The mixed boundary system is solved as:

\mathbf{M}\mathbf{j} = \mathbf{h} \tag{16.17}

where \mathbf{M} is assembled row-by-row from \mathbf{C} (for known sources) and \mathbf{D} (for known emissive powers), and \mathbf{j} is the total radiant power vector.

16.6 Advantages of the Anisotropic Extension

Physical Fidelity: Captures directional dependence of reflection-scattering, enabling more accurate modeling of real materials and surfaces.
Analytical Framework: Maintains the analytical structure of the exchange factor transformation, avoiding iterative Monte Carlo methods.
Backward Compatibility: Reduces exactly to the isotropic case when \mathbf{\Gamma}_{ij} = 0.5 for all elements (or c = 0), ensuring consistency with established methods.
Preserved Properties: Non-negativity (\mathbf{K} \geq 0) and physical bounds (\mathbf{K} \leq \mathbf{F}) are guaranteed by proper choice of the scaling coefficient c (see Chapter Chapter 19 for proofs).
Maximum Flexibility: The analytical bounds on c provide the maximum possible anisotropic variation while maintaining physical validity.
Element-Wise Control: Full N \times N degrees of freedom in \mathbf{B}_{\mathrm{init}} allow each incident-receiver pair to have independent anisotropic behavior.

16.7 Physical Applications

For surfaces with specular components, media with anisotropic scattering phase functions (e.g., Mie scattering, Henyey-Greenstein, i.e. (ij,k)-three-index dependencies) and surfaces with oriented microstructure (e.g., brushed metal, fibrous materials), more complex models are required.

16.8 Conclusion

The anisotropic extension of the exchange factor transformation provides a mathematically rigorous and physically meaningful framework for modeling directional reflection-scattering in radiative transfer. By introducing the anisotropy matrix \mathbf{\Gamma} with the bounded perturbation parameterization, and employing row-stochastic normalization, the method preserves all essential properties while significantly expanding the range of physical phenomena that can be accurately modeled.

# Anisotropic Extension of the Exchange Factor Transformation ## Introduction The exchange factor transformation described in Chapter @sec-exchange_fact_trans enables analytical treatment of multiple reflection-scattering processes in radiative transfer by transforming the exchange factor matrix $\mathbf{F}$ into absorption and reflection-scattering matrices $\mathbf{A}$ and $\mathbf{R}$. The method assumes isotropic reflection-scattering behavior, where the probability of reflection-scattering from element $j$ is independent of the incident element $i$. This is represented by a column-constant single reflection-scattering matrix $\mathbf{B}$. However, many physical systems exhibit anisotropic reflection-scattering, where the probability and directional distribution of reflection-scattering depend on the incident direction. This chapter presents an extension of the exchange factor transformation that accommodates anisotropic reflection-scattering while preserving all fundamental properties: non-negativity, energy conservation, and analytical solvability. ## Anisotropic Reflection-Scattering Matrix The anisotropic extension introduces a full matrix $\mathbf{\Gamma}$ (Gamma) to characterize directional dependence of reflection-scattering: $$ \mathbf{\Gamma} \in \mathbb{R}^{N \times N}, \quad 0 \leq \mathbf{\Gamma}_{ij} \leq 1 \quad \forall \; ij $$ {#eq-gamma-bounds} where $\mathbf{\Gamma}_{ij}$ represents the anisotropy parameter controlling how incident radiation from element $i$ affects reflection-scattering from element $j$. ### Physical Interpretation The matrix $\mathbf{\Gamma}$ provides directional control over reflection-scattering behavior: - $\mathbf{\Gamma}_{ij} = 0.5$: Baseline (isotropic) reflection-scattering from $j$ due to incidence from $i$ - $\mathbf{\Gamma}_{ij} < 0.5$: Reduced reflection-scattering from $j$ due to incidence from $i$ (directional suppression) - $\mathbf{\Gamma}_{ij} > 0.5$: Enhanced reflection-scattering from $j$ due to incidence from $i$ (directional enhancement) When $\mathbf{\Gamma}_{ij} = 0.5$ for all elements (or equivalently when the scaling coefficient $c = 0$), the anisotropic formulation reduces exactly to the isotropic case with column-constant reflection-scattering matrix $\mathbf{B}$. ## Construction of the Anisotropic Interaction Matrix The anisotropic interaction-reflection-scattering matrix $\mathbf{K}$ is constructed through a multi-step process that ensures physical consistency. **Step 1:** Initial Reflection-Scattering Distribution Define the initial reflection-scattering distribution matrix: $$ \mathbf{B}_{\mathrm{init},ij} = \mathbf{b}_j + c(2\mathbf{\Gamma}_{ij} - 1)\min(\mathbf{b}_j, 1-\mathbf{b}_j) $$ {#eq-b-init} where: - $\mathbf{b}_j$ is the baseline reflection-scattering coefficient of element $j$ - $\mathbf{\Gamma}_{ij} \in [0,1]$ controls directional anisotropy - $c > 0$ is a scaling coefficient that determines the strength of anisotropic perturbations - $\min(\mathbf{b}_j, 1-\mathbf{b}_j)$ limits the perturbation range to maintain physical validity This parameterization maps: - $\mathbf{\Gamma}_{ij} = 0 \rightarrow$ maximum downward perturbation: $\mathbf{B}_{\mathrm{init},ij} = \mathbf{b}_j - c\min(\mathbf{b}_j, 1-\mathbf{b}_j)$ - $\mathbf{\Gamma}_{ij} = 0.5 \rightarrow$ baseline (isotropic): $\mathbf{B}_{\mathrm{init},ij} = \mathbf{b}_j$ - $\mathbf{\Gamma}_{ij} = 1 \rightarrow$ maximum upward perturbation: $\mathbf{B}_{\mathrm{init},ij} = \mathbf{b}_j + c\min(\mathbf{b}_j, 1-\mathbf{b}_j)$ The coefficient $c$ must be chosen to guarantee that the final matrix $\mathbf{K}$ satisfies $0 \leq \mathbf{K}_{ij} \leq \mathbf{F}_{ij}$ for all elements. The maximum safe value $c_{\max}$ can be computed analytically from the constraint equations (see Chapter @sec-exchange_fact_anisotropic_extend_analysis for derivation). The maximum safe coefficient is: $$ c_{\max} = \min\left(c_{\max,\text{nonneg}}, c_{\max,\text{leq F}}\right) $$ where: $$ c_{\max,\text{nonneg}} = \min_{i,j : \text{denom} > 0} \frac{b_j}{\Delta_i(1-b_j) - (2\Gamma_{ij}-1)m_j} $$ $$ c_{\max,\text{leq F}} = \min_{i,j : \text{denom} > 0} \frac{1-b_j}{(2\Gamma_{ij}-1)m_j - \Delta_i(2-b_j)} $$ For numerical safety, a lower value is used: $c = \alpha \cdot c_{\max}$ with $\alpha < 1$. This guarantees $0 \leq \mathbf{K}_{ij} \leq \mathbf{F}_{ij}$ for all elements. ::: {.callout-important} ## Key Properties 1. **Negative B allowed**: $\mathbf{B}_{\mathrm{init},ij}$ can be negative for sufficiently large $c$, but the transformation guarantees $\mathbf{K}_{ij} \geq 0$. 2. **Element-wise freedom**: Full $N \times N$ degrees of freedom in $\mathbf{B}_{\mathrm{init}}$. 3. **Optimal bounds**: Since the bounds on $c$ were derived analytically they provide maximum flexibility for any given $\mathbf{\Gamma}$. ::: **Step 2:** Interaction-Reflection-Scattering (Pre-Normalization) Compute the element-wise product of the exchange factor matrix and the initial reflection-scattering distribution: $$ \mathbf{K}_{\mathrm{init},ij} = \mathbf{F}_{ij} \cdot \mathbf{B}_{\mathrm{init},ij} $$ {#eq-k-init} This represents the probability that radiation emitted from element $i$ interacts with element $j$ and is then reflected-scattered, before normalization. **Step 3:** Enforce Probability Conservation Create a matrix that accounts for both reflection-scattering and absorption: $$ \mathbf{U}_{ij} = \mathbf{K}_{\mathrm{init},ij} + \mathbf{F}_{ij} \cdot \mathbf{P}_{jj} $$ {#eq-enforce-unity} Each emitted ray either reflects-scatters ($\mathbf{K}_{\mathrm{init}}$) or is absorbed ($\mathbf{F} \cdot \mathbf{P}$). **Step 4:** Compute Row Sums Calculate the row sums: $$ \mathbf{s}_i = \sum_j \mathbf{U}_{ij} = \sum_j (\mathbf{K}_{\mathrm{init},ij} + \mathbf{F}_{ij} \cdot \mathbf{P}_{jj}) $$ {#eq-row-sums} **Step 5:** Normalize to Row-Stochastic Normalize the matrix to ensure row sums equal unity: $$ \mathbf{T}_{ij} = \frac{\mathbf{U}_{ij}}{\mathbf{s}_i} $$ {#eq-normalize} This creates a row-stochastic matrix, ensuring that total probability is conserved for radiation emitted from each element $i$. **Step 6:** Final Interaction-Reflection-Scattering Matrix Extract the reflection-scattering component: $$ \mathbf{K}_{ij} = \mathbf{T}_{ij} - \mathbf{F}_{ij} \cdot \mathbf{P}_{jj} $$ {#eq-k-final} Substituting from @eq-normalize: $$ \mathbf{K}_{ij} = \frac{\mathbf{K}_{\mathrm{init},ij} + \mathbf{F}_{ij} \cdot \mathbf{P}_{jj}}{\mathbf{s}_i} - \mathbf{F}_{ij} \cdot \mathbf{P}_{jj} $$ {#eq-k-explicit} Simplifying: $$ \mathbf{K}_{ij} = \frac{\mathbf{K}_{\mathrm{init},ij}}{\mathbf{s}_i} - \mathbf{F}_{ij} \cdot \mathbf{P}_{jj} \left(1 - \frac{1}{\mathbf{s}_i}\right) $$ {#eq-k-simplified} ## Properties of the Row Sum Normalization Factor ### Bounds on s From @eq-row-sums: $$ \mathbf{s}_i = \sum_j (\mathbf{K}_{\mathrm{init},ij} + \mathbf{F}_{ij} \cdot \mathbf{P}_{jj}) $$ Since $\mathbf{K}_{\mathrm{init},ij} = \mathbf{F}_{ij} \cdot \mathbf{B}_{\mathrm{init},ij}$, we have: $$ \mathbf{s}_i = \sum_j \mathbf{F}_{ij} \cdot (\mathbf{B}_{\mathrm{init},ij} + \mathbf{P}_{jj}) $$ {#eq-s-expanded} Since $\mathbf{F}$ is row-stochastic ($\sum_j \mathbf{F}_{ij} = 1$), $\mathbf{F}_{ij} \geq 0$, and $\mathbf{P}_{jj} = 1-\mathbf{b}_j > 0$: $$ \mathbf{s}_i > 0 $$ {#eq-s-lower-bound} The normalization factor $\mathbf{s}_i$ is always positive, preventing division by zero in equations @eq-k-explicit and @eq-k-simplified. ### Special Case: Isotropic Limit When $\mathbf{\Gamma}_{ij} = 0.5$ for all $i,j$ (or equivalently when $c = 0$), we have $\mathbf{B}_{\mathrm{init},ij} = \mathbf{b}_j$ (column-constant): $$ \mathbf{s}_i = \sum_j \mathbf{F}_{ij} \cdot (\mathbf{b}_j + (1-\mathbf{b}_j)) = \sum_j \mathbf{F}_{ij} = 1 $$ Therefore, the normalization does nothing, and: $$ \mathbf{K}_{ij} = \mathbf{F}_{ij} \cdot \mathbf{b}_j $$ This recovers the isotropic case exactly: $\mathbf{K} = \mathbf{F} \circ \mathbf{B}$, where $\mathbf{B}$ is the column-constant matrix of reflection-scattering coefficients. ## Subsequent Transformation Once $\mathbf{K}$ is constructed, the remainder of the exchange factor transformation proceeds almost identically to the isotropic case: The steady-state path matrix is given by: $$ \mathbf{S}_\infty = (\mathbf{I} - \mathbf{K})^{-1} \mathbf{F} $$ {#eq-s-infinity} The absorption and reflection-scattering matrices are given by the alternate formulation: $$ \mathbf{A} = \mathbf{P}(\mathbf{I}-\mathbf{K})^{-1}(\mathbf{F}-\mathbf{K}) $$ {#eq-a-matrix} $$ \mathbf{R} = \mathbf{P}(\mathbf{I}-\mathbf{K})^{-1}\mathbf{K} $$ {#eq-r-matrix} The system matrice $\mathbf{C}$ and $\mathbf{D}$ are given by: $$ \mathbf{C} = \mathbf{I} - \mathbf{A}^T - \mathbf{R}^T $$ {#eq-c-matrix} $$ \mathbf{D} = \mathbf{I} - \mathbf{R}^T $$ {#eq-d-matrix} The mixed boundary system is solved as: $$ \mathbf{M}\mathbf{j} = \mathbf{h} $$ {#eq-system} where $\mathbf{M}$ is assembled row-by-row from $\mathbf{C}$ (for known sources) and $\mathbf{D}$ (for known emissive powers), and $\mathbf{j}$ is the total radiant power vector. ## Advantages of the Anisotropic Extension 1. **Physical Fidelity**: Captures directional dependence of reflection-scattering, enabling more accurate modeling of real materials and surfaces. 2. **Analytical Framework**: Maintains the analytical structure of the exchange factor transformation, avoiding iterative Monte Carlo methods. 3. **Backward Compatibility**: Reduces exactly to the isotropic case when $\mathbf{\Gamma}_{ij} = 0.5$ for all elements (or $c = 0$), ensuring consistency with established methods. 4. **Preserved Properties**: Non-negativity ($\mathbf{K} \geq 0$) and physical bounds ($\mathbf{K} \leq \mathbf{F}$) are guaranteed by proper choice of the scaling coefficient $c$ (see Chapter @sec-exchange_fact_anisotropic_extend_analysis for proofs). 5. **Maximum Flexibility**: The analytical bounds on $c$ provide the maximum possible anisotropic variation while maintaining physical validity. 6. **Element-Wise Control**: Full $N \times N$ degrees of freedom in $\mathbf{B}_{\mathrm{init}}$ allow each incident-receiver pair to have independent anisotropic behavior. ## Physical Applications For surfaces with specular components, media with anisotropic scattering phase functions (e.g., Mie scattering, Henyey-Greenstein, i.e. (ij,k)-three-index dependencies) and surfaces with oriented microstructure (e.g., brushed metal, fibrous materials), more complex models are required. ## Conclusion The anisotropic extension of the exchange factor transformation provides a mathematically rigorous and physically meaningful framework for modeling directional reflection-scattering in radiative transfer. By introducing the anisotropy matrix $\mathbf{\Gamma}$ with the bounded perturbation parameterization, and employing row-stochastic normalization, the method preserves all essential properties while significantly expanding the range of physical phenomena that can be accurately modeled.